401
Order 6~ 401408, 1990.
0 WQ Kluwer Acaakmic Publishers. Printed in the Netherlands.
Duality and Finite Spaces
JUAN ANTONIO
NAVARRO
GONZALEZ
Departamento aYeMatemdticas, Universidad a2 Extremadura, 04Wl-Badajoz,
Spain.
Communicated by K. Keimel
(Received: 4 May 1988; accepted: 30 December 1989)
Abstract. Using that finite topological spaces are just finite orders, we develop a duality theory for
sheaves of Abelian groups over finite spaces following closely Grothendieck’s duality theory for coherent
sheaves over proper schemes. Since the geometric realization of a finite space is a polyhedron, we relate
this duality with the duality theory for AbeIian sheaves over polyhedra.
AMS subject clas4fications (1985 revision): 55MO5, 55N30, 06AlO.
Key work
Finite spaces, sheaf cohomology, dualizing complex, geometric realization.
0. Introduction
Finite topological spaces have increasing importance in mathematics. Since they are
just finite posets, they correspond with finite distributive lattices; hence their
combinatorial importance is clear, as well as in lattice theory. Moreover, each finite
space X has a geometric realization 1x1 which is a triangulated polyhedron and so
we get any polyhedron [ 1, 121. There is a canonical map 1x1+X and it is a
homotopical equivalence [ 111, so that finite spaces include the homotopical study of
triangulated polyhedra and show why classical algebraic topology is essentially of
combinatorial nature. Finite spaces have been used in studies on foundations of
Homotopy Theory [ 71. On the other hand, coherent spaces (spectrums of distributive
lattices) are just projective limits of finite spaces and therefore finite spaces provide
a cornerstone for the development of a dimension theory for arbitrary topological
spaces and locales [9, 131. Finally, the topological space underlying any Noetherian
n-dimensional scheme is a projective limit of n-dimensional finite spaces.
Finite spaces provide a unifying tool in mathematics, relating in a systematic way
different areas where combinatorial ideas play a role.
In this paper we develop a duality theory for sheaves of Abelian groups on finite
spaces, following closely Grothendieck’s duality theory for coherent sheaves on
proper schemes ([4]). Since the geometric realization 1x1 of any finite space X is a
polyhedron, we have Verdier’s duality theory for sheaves of Abelian groups on 1x1.
We prove that the canonical map Ix~+ X induces a quasi-isomorphism between their
respective dualizing complexes.
In this paper any topological space is assumed to be T,-,.
402
JUAN ANTONIO
NAVARRO
GONZALEZ
1. Finite Topological Spaces
DEFINITION
1.1. A finite space is a T&opological
points, i.e. with a finite number of closed subsets.
space with a finite number of
It is well-known [lo] that a finite space is the same as a finite poset (short for
partially ordered set). If X is a finite poset, then we consider a topology on X by
letting all hereditary subsets (subsets Z of X such that any point preceding a point
of Z belongs to Z) be the closed sets. This topology determines the order, since we
have .x <y just when x lies in the closure of y. Conversely, any T,,-topology on a
finite set defines an order on it: x 6 y if and only if x is in the closure of JJ. This
order determines the given topology because, the set being finite, it is determined
once we know the closure of each point. We shall always identify finite spaces with
finite posets, using each time the most convenient term for a clear understanding. It
is obvious that continuous maps correspond with order-preserving maps (x <y
implies f(x) <f(y)).
1.1. HOMOLOGY
Now, to fix ideas and notation, we shall briefly recall some well-known facts about
the homology of finite posets [3].
If X is a finite poset, we denote by C,,(X, Z) the Abelian free group generated by
sequences x0 < x1 < * * * < X~ of points of X. We define morphisms d: C,,(X, Z) +
Cn - r (X, Z) for n 2 1 by defining them on the generators:
It is not difficult to show that d* = 0, hence we have defined a chain complex
C.(X, Z) which is called the chain complex of X. Given an Abelian group G, the
chain complex of X with coefficients G is defined to be C.(X, G) = C.(X, Z) @G.
The n-th /zonzo/ogy group of X with coefficients G is the n-th homology group of
the chain complex C.(X, G) and it is denoted by H,,(X, G). It is clear that C.(X, G)
and Hn(X, G) are both natural in the variables X and G.
Let X, Y be two finite posets. Then the set Hom(X, Y) of all order-preserving
maps X+ Y has a canonical order:
f <g
if and only if f(x) < g(x)
for any point x of X
so that Hom(X, Y) is also a finite space. We say that two continuous maps are
homotopic when they are in the same connected component of Hom(X, Y). It is
clear that any finite poset with a unique minimal point (or maximal point) is a
contractible space: it has the homotopy type of a point.
Moreover, the morphism f*: Hn(X, G) + H,,( Y, G) only depends on the homotopy class of the continuous map f: X -+ Y.
DUALITY
AND FINITE SPACES
403
1.2. COHOMOLOGY
Any finite space is, by definition, a topological space. Hence, we have at our disposal
the cohomology theory of Abelian sheaves on arbitrary topological spaces due to
Grothendieck [6]. Now we recall the most basic facts about it [ $61:
A sheuf F of Abelian groups on a topological space X assigns to every open subset
U z X an Abelian group F(U) and to every inclusion map U s V a morphism
rUV: F(V) -+ F(U) so that
(11F(4)= a
G9 rUw = ruVrVW
whenever U s V G W.
(3) Given an open cover {U,.} of an open subset U and a family of elements
si E F(Ui) such that for each pair (i,j) we haver r&) = r(.+) in F(Ui n Uj),
there exists a unique s E F(U) with si = r(s) for a11 index i.
As U varies over all open neighbourhoods of a fixed point x of X, the collection
F(U) is a direct system of Abelian groups and the sfulk of X at x is defined to be
Fx = lim F(U).
XZIJ
A sequence of sheaves of Abelian groups is exact if and only if it is exact on stalks.
Given a topological space X, the cohomology functors ZP(X, -) are defined to
be the right derived functors of the additive functor F ,SVF(X). This makes sense
because F P.VF(X) is left exact and the category of sheaves of Abelian groups on X
has enough injectives [6]. These cohomology groups have the genera1 properties of
derived functors (long exact cohomology sequence, spectra1 sequences, etc.). For any
sheaf of Abelian groups F on X, the group EP’(X, F) is said to be the p-th cohomoZogy
group of X with coefficients F.
Let f: X-S Y be a continuous map of topological spaces. For any sheaf F of
Abelian groups on X, the direct imuge f * F is defined by (f a F)( V) = F( f - ‘( V)) for
any open set V s Y. If G is a sheaf of Abelian groups on Y, we may define the inverse
image FG since the functor f* has a left adjoint f’?
Hom#+G,
F) = HomrJG,f*F).
It is well-known that we have (pG)X = G,(x) for any point x E X.
The higher direct image functors RPf* are defined to be the right derived functors
of the direct image functor f*.
When f is the projection of X onto a point, we have F(X) = f * F, and f+G is said
to be the corzsrurzfsheuf G on X. Hence, in this case Rpf*(F) = IP(X, F).
These results hold for arbitrary topological spaces, but we are interested in a very
particular case: when X is a finite space. If X is finite, each point x has a minima1
neighbourhood Ux, so that Fx = F(UJ. Hence, a sheaf F of Abelian groups on a
finite. poset X is just a family of Abelian groups {Fx}, x E X, and morphisms
ryx . Fx + F,,, x < y, such that rz,,r,,x = rzX whenever x < y < z (cf. the linear repre-
JUAN ANTONIO
404
NAVARRO
GONZALEZ
sentations of X, [ 151). For example, the constant sheaf G is just Fx = G for any
point x and rl,, = idG for any pair x < JJ.
When X is a finite space, the cohomology groups P’(X, F) have the following
combinatorial interpretation:
Let F be a sheaf of Abelian groups on a finite poset X. Then we denote by
C”(X, F) the Abelian group
WK
Fl = xo<-j<x
. .”
En
and we define morphisms d: C’(X, F) + C” + ‘(X, F) by the following formula:
=~<~<~(-l)~~(x~~...~,...~x~+,)+(-l)~+’~(x~~...~x~)
. .
where Z(x,,<***<xJ
is the image of u(xO < * * +< XJ under the morphism,
t. : En + Fxn + , - It is easy to check that 0 = d*, so that we have a cochain complex
C-(X, F) (see [3]). Now, we have:
THEOREM
1.2 [8]. ZP(X, F) = Hp[C-(X, F)].
COROLLARY
1.3. If G is an Abelian group, the groups Hp(X, G) are isomorphic to
the cohomology groups of the complex Homz(C.(X, Z), G).
COROLLARY
1.4. Constant sheaves over contractible jinite spaces are acyclic.
Finally, we shall need the following result:
DEFINITION
1.5. We define the dimension of a finite poset to be the supremum
of all integers n such that there exists a properly ascending chain x0 < x, < * . . < X~
of points of X.
THEOREM
1.6 [6]. The cohomology groups HP(X, F) vunish when p is greuter thun
the dimension of the jinite space X.
2. Duality
Now we develop a duality theory for sheaves of Abelian groups on finite spaces,
following
Verdier’s
If F is
by C*(F)
closely Grothendieck’s work [4] for proper maps between schemes and
work [ 141 for classical topological spaces.
a sheaf of Abelian groups over a finite space X of dimension n, we denote
the n-th truncation of the canonical flasque resolution of F [5]:
C’(F) = Co(F) A
C’(F) . . . Cn - *(F) ?!+
C?-‘(F)
+Coker
If x’ is a complex of sheaves over X, we denote by C*(S.)
complex associated to the double complex { CJ’(xq)}.
dnPl.
the singly graded
DUALITY
AND FINITE
405
SPACES
The duality theory for a continuous map
simple because of the following facts:
f: X + S between finite spaces is very
(a) The stalk of Pf*(F) at any point p of S is W( f- ‘(Up), F), where UP is the
minimal neighborhood of p in S. Hence, it vanishes when i is greater than the
dimension of X (1.5).
(b) The functor f* commutes with filtered inductive limits, since finite spaces are
obviously Noetherian ([ 51, 113,lO. 1).
(c) The functorial resolution C’(F) is bounded, f*-acyclic and commutes with
direct sums and filtered inductive limits, because any direct product which
appear in the definition of the canonical flasque resolution is finite when X is
finite.
DUALITY
THEOREM 2.1. Let f: X + S be a continuous map betweenfinite spaces
and let I’ be an injective resolution of the constant sheaf Z on S. There is a complex
D> of injective sheaves over X such that, for every complex x’ of sheaves of Abelian
groups over X we have a functorial isomorphism
Hom’(f*(C’(&-‘))
r) = Hom’(.%‘, D;)
where Horn* &notes the complex of morphisms. Moreover, D> is bounded below.
Proof Fix two integer numbers p, q and let us consider the contravariant functor
which assigns to each sheaf F of Albelian groups over X the Abelian group
Hom( f*(Cj’(F), Zq)). This functor is exact (because G’(F) is f*-acyclic and Zq is an
injective sheaf) and it takes inductive limits into projective limits; hence it is
representable [4]. So there is an injective sheaf D -KWon X and a natural isomorphism
Hom( f*(C?‘(F),
Zq)) = Hom(F, D -J’vq).
Now let Dy = @ i +j = d DiJ. For each integer number n we have
Homn(f*(C*(.%‘)),
Z’) = n Horn
1
= fl
n
i
p+q=i
=n
n
q
P+q=l
Z
@
Hom(f*(CJ’($-q)),
Horn(xq,
D-Psi+“)
om xq,p+T=i
D-Pvi+”
q
rIH
Hom’(f*(C’(x’)),
is an isomorphism
isomorphism.
r) = Hom*(%‘,
of complexes.
Zi+”
Zi+“)
J
c
= Homn(s’,
and this is a functorial
degree 1 such that
fe(Cp(Xq)),
+q=i
D; )
Hence there is a differential
D;)
d: Dj-+ D; of
406
JUAN ANTONIO
NAVARRO
GONZALEZ
Remarks. If the dimension of X is n, then Df = 0 for p < -n.
Let P be the one point space and let f: X + P. We define the dualizing complex
D> of X to be D;.
The complex D> depends on the given resolution Z. If we choose different
injective resolutions of the constant sheaf 2!, then we get quasi-isomorphic
plexes. Hence the complex D; is well-defined up to quasi-isomorphisms.
com-
Note. The above argument may be applied to any bounded below complex Z* of
injective sheaves on S, so that we get a bounded below complex f’(Z’) of injective
sheaves on X such that Hom’(.?K’, f'(r)) = Horn-( f*(C’(&-*)),
r). Hence we have
a functor f ‘: D +(S) + D +(X), between the respective derived categories of bounded
below complexes of sheaves of Abelian groups, which is a right-adjoint of the
functor RF*: D(X) + D(S):
R Horn-($-‘, f!(L’))
COROLLARY
= R Hom’(Rf*($-‘),
Z’).
2.2. Zf F is a sheaf of Abelian groups on X, then there is a split exact
sequence
0 + Ext$(ZP+ ‘(X, F), 2?)+ Extei(F, D>) + Hom(ZZ’(X, F), J!) + 0.
ProoJ If Z denotes an injective resolution of the group 2!, then there is a split
exact sequence
0 + Exti(Hi+
‘(X, F), 2?) + ZP[Hom*(r(C’(F)),
Z)] + Hom(ZP(X, F), i?) + 0.
Now, by the Duality Theorem we have Hom’(I(C’(F),
Z) = Hom’(F, D>) and
ZPi[Hom*(F, D>)] is just Ext-‘(F, D>) because Di is a complex of injective
sheaves.
THEOREM
2.3. The homology groups of a$nite space are just the hypercohomology
groups of its dualizing complex:
Hi(X, Z) = Wi(X,
Di).
ProojI Let Z be an injective resolution of the group z. Then
l-(X, D>) = Hom’(&
D>) = Hom’(I(X,
C’(z)), Z)
and, by ( 1.8) we have that the natural morphisms
are quasi-isomorphisms (recall that C*(X, G) denotes the cochain complex with
coefficients G). Therefore, the complex I(X, Di ) is quasi-isomorphic
to
Hom’(C*(X, Q, Z’) and this complex is quasi-isomorphic to Hom’(C*(X, Q, 2!)
since C-(X, i2) is a complex of free Abelian groups.
Now, it is clear that Hom.(C*(X, 2!)) = C.(X, 2!) and we conclude that
W’(X, D>) = H-‘[Hom’(C’(X,
22) 2!)] is ZZi(X, Q.
DUALITY
407
AND FINITE SPACES
3. Geometric Realization
Given a finite poset X, we may define a simphcial complex C(X) on the vertex set
X by letting all finite chains be the faces. The geometric realization of C(X) will be
denoted by 1x1 and it is said to be the geometric reulizution of X (see [ 11, [ 111 and
[ 121). There is a functorial continuous map
which sends each point of 1x1 to the greatest element in its carrier chain. This map
is a homotopy equivalence [ 111,
THEOREM
3.1. Zf F is a sheaf of Abelian groups on a jinite space X, the inverse
image
g*: HP(X, F) -+ HP(lXl, g*F)
is an isomorphism for all p > 0.
ProoJ It is a well-known result for constant sheaves. So it is also true when
F = ZA, for any closed subset A s X, because g-‘(A)
is just [A]. The exact
cohomology
sequences associated to the following
short exact sequences
prove that it is also true when F = ZX- A (see [ 51 for a definition of these sheaves).
Finally, remark that the theorem is valid for a direct sum whenever it holds for each
summand.
Now, let F be an arbitrary sheaf of Abelian groups on X. By ([5], II 2.9.2), F is
a quotient of a direct sum of sheaves ZU, so that F is quasi-isomorphic to a
bounded above complex X’ such that the theorem holds for each sheaf Xp. Since
we have the following convergent spectral sequences
ET9 = Hp(X, X9) =c.HP+ 9(X, F)
,??qq= HJ’([X[, g*(.U))
* HJ’+q(lX[, g*F)
and g* : Eyq + i@q is an isomorphism
for
g*: Hn(X, F) + Hn(lX[, g*F) is an isomorphism.
any p and
q, we
get that
(3.2) The Duality Theorem 2.2 holds for any continuous map between finite
polyhedra (in fact it holds under much more general hypotheses, see [ 141). Hence,
if P is a finite polyhedron, we have its dualizing complex D;. Now we compare Di
and DiA when X is a finite space:
THEOREM 3.3. D> und g*(Diq) ure quusi-isomorphic.
ProojI Let r be an injective resolution of the gorup Z. For any sheaf F of
Abelian groups on X we have
408
JUAN ANTONIO
Hom*(F, g*(Diq))
= Hom*(g*F,
NAVARRO
GONZALEZ
Diq)
= Horn-( l-( 1x1, C’(g*F)),
and, by (3.1), this complex is quasi-isomorphic
r)
to
Hom’( l-(X, C’(F)), 1;) = Hom*(F, D> ).
Since it holds for an arbitrary sheaf F, we may conclude that D; is quasi-isomorphic to g*Diq.
Acknowledgements
These results were proposed to the author some years ago by Prof. Juan Sancho
GuimerA. The author is very grateful to him for his constant advice and help.
Thanks are due to the referee for having proposed the clear statement of Theorems
3.1 and 3.3.
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